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Properties

Label 2-140-140.123-c1-0-9
Degree $2$
Conductor $140$
Sign $0.287 - 0.957i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.24 + 0.667i)2-s + (0.543 + 2.02i)3-s + (1.10 + 1.66i)4-s + (−1.62 − 1.53i)5-s + (−0.676 + 2.89i)6-s + (0.0742 − 2.64i)7-s + (0.270 + 2.81i)8-s + (−1.22 + 0.704i)9-s + (−0.998 − 3.00i)10-s + (−0.366 − 0.211i)11-s + (−2.77 + 3.15i)12-s + (−1.56 − 1.56i)13-s + (1.85 − 3.24i)14-s + (2.23 − 4.12i)15-s + (−1.54 + 3.69i)16-s + (1.18 + 4.41i)17-s + ⋯
L(s)  = 1  + (0.881 + 0.472i)2-s + (0.313 + 1.17i)3-s + (0.554 + 0.832i)4-s + (−0.726 − 0.687i)5-s + (−0.276 + 1.18i)6-s + (0.0280 − 0.999i)7-s + (0.0955 + 0.995i)8-s + (−0.406 + 0.234i)9-s + (−0.315 − 0.948i)10-s + (−0.110 − 0.0637i)11-s + (−0.800 + 0.910i)12-s + (−0.433 − 0.433i)13-s + (0.496 − 0.867i)14-s + (0.576 − 1.06i)15-s + (−0.385 + 0.922i)16-s + (0.286 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.287 - 0.957i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37107 + 1.01976i\)
\(L(\frac12)\) \(\approx\) \(1.37107 + 1.01976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.667i)T \)
5 \( 1 + (1.62 + 1.53i)T \)
7 \( 1 + (-0.0742 + 2.64i)T \)
good3 \( 1 + (-0.543 - 2.02i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.366 + 0.211i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.56 + 1.56i)T + 13iT^{2} \)
17 \( 1 + (-1.18 - 4.41i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.66 + 6.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.44 - 1.72i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.72iT - 29T^{2} \)
31 \( 1 + (1.70 + 0.982i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.20 + 1.92i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.01T + 41T^{2} \)
43 \( 1 + (1.88 - 1.88i)T - 43iT^{2} \)
47 \( 1 + (1.70 - 6.35i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.780 + 0.209i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.35 - 2.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.925 - 1.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.70 + 2.33i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.16iT - 71T^{2} \)
73 \( 1 + (-6.67 + 1.78i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.77 + 3.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.71 - 4.71i)T - 83iT^{2} \)
89 \( 1 + (-10.0 + 5.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.64 - 4.64i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.31562706417711470780867531011, −12.68156628493962629126933277967, −11.29358330895802962634230681776, −10.49872986811783067465685508870, −9.069377614892846305909687187146, −8.029005466855319978346869614179, −6.91667241112172302872601660451, −5.10028010858703637515685481301, −4.32371982507303622854007394598, −3.38699846775626402746226689821, 2.06187929405194383416754488374, 3.26460481896013621210120949027, 5.05315663264539663393835397885, 6.55215165534978628404351052001, 7.26797003528174439363177317574, 8.583791375396942989005802545987, 10.15919794136858769736891685504, 11.36953663532449475221783387207, 12.22957627480421800394328613503, 12.67186748888065243201756268388

Graph of the $Z$-function along the critical line